Browse by

Terminology of Languages

 Alphabet : a finite set of symbols (ASCII characters)  String :
± ± ± ± ± ± ± ± Finite sequence of symbols on an alphabet Sentence and word are also used in terms of string I is the empty string |s| is the length of string s.  the empty set is a language. {I} the set containing empty string is a language The set of well-formed C programs is a language The set of all possible identifiers is a language. Is=s

Regular Expressions
 We use regular expressions to describe tokens of a programming language.  A regular expression is built up of simpler regular expressions (using defining rules)  Each regular expression denotes a language.  A language denoted by a regular expression is called as a regular set.

Regular Definitions
 To write regular expression for some languages can be difficult, because their regular expressions can be quite complex. In those cases, we may use regular definitions.  We can give names to regular expressions, and we can use these names as symbols to define other regular expressions.  A regular definition is a sequence of the definitions of the form: d1 p r1 where di is a distinct name and d2 p r2 ri is a regular expression over symbols in . 7{d1,d2,...,di-1} dn p rn
basic symbols previously defined names
13

Finite Automata
 A recognizer for a language is a program that takes a string x, and answers ³yes´ if x is a sentence of that language, and ³no´ otherwise.  We call the recognizer of the tokens as a finite automaton.  A finite automaton can be: deterministic(DFA) or non-deterministic (NFA)  This means that we may use a deterministic or non-deterministic automaton as a lexical analyzer.  Both deterministic and non-deterministic finite automaton recognize regular sets.  Which one?
± deterministic ± faster recognizer, but it may take more space ± non-deterministic ± slower, but it may take less space ± Deterministic automatons are widely used lexical analyzers.

 First, we define regular expressions for tokens; Then we convert them into a DFA to get a lexical analyzer for our tokens.
± Algorithm1: Regular Expression ± Algorithm2: Regular Expression NFA DFA (two steps: first to NFA, then to DFA) DFA (directly convert a regular expression into a DFA)
15

Non-Deterministic Finite Automaton (NFA)
 A non-deterministic finite automaton (NFA) is a mathematical model that consists of:
± ± ± ± ± S - a set of states 7 - a set of input symbols (alphabet) move ± a transition function move to map state-symbol pairs to sets of states. s0 - a start (initial) state F ± a set of accepting states (final states)

 I- transitions are allowed in NFAs. In other words, we can move from one state to another one without consuming any symbol.  A NFA accepts a string x, if and only if there is a path from the starting state to one of accepting states such that edge labels along this path spell out x.
16

NFA (Example)
a 0 b a 1 b 2

start

Transition graph of the NFA

0 is the start state s0 {2} is the set of final states F 7 = {a,b} S = {0,1,2} Transition Function: a 0 {0,1} 1 _ 2 _

b {0} {2} _

The language recognized by this NFA is (a|b) * a b

17

Deterministic Finite Automaton (DFA)
 A Deterministic Finite Automaton (DFA) is a special form of a NFA.
 no state has I- transition

 for each symbol a and state s, there is at most one labeled edge a leaving s. i.e. transition function is from pair of state-symbol to state (not set of states)

b 0 a

a 1 b

a b

The language recognized by
2

this DFA is also (a|b) * a b

18

Implementing a DFA
 Le us assume that the end of a string is marked with a special symbol (say eos). The algorithm for recognition will be as follows: (an efficient implementation)
s s0 c nextchar while (c != eos) do begin s move(s,c) c nextchar end if (s in F) then return ³yes´ else return ³no´ { start from the initial state } { get the next character from the input string } { do until the en dof the string } { transition function }

{ if s is an accepting state }

19

Implementing a NFA
S I-closure({s0})
c nextchar while (c != eos) { begin s I-closure(move(S,c)) c nextchar end if (SF != *) then return ³yes´ else return ³no´ { if S contains an accepting state } { set of all states can be accessible from a state in S by a transition on c } { set all of states can be accessible from s0 by I-transitions }



This algorithm is not efficient.
20

Converting A Regular Expression into A NFA (Thomson¶s Construction)
 This is one way to convert a regular expression into a NFA.  There can be other ways (much efficient) for the conversion.  Thomson¶s Construction is simple and systematic method. It guarantees that the resulting NFA will have exactly one final state, and one start state.  Construction starts from simplest parts (alphabet symbols). To create a NFA for a complex regular expression, NFAs of its sub-expressions are combined to create its NFA,

21

Thomson¶s Construction (cont.)
I

 To recognize an empty string I  To recognize a symbol a in the alphabet 7

i

f

i

a

f

 If N(r1) and N(r2) are NFAs for regular expressions r1 and r2  For regular expression r1 | r2

Thomson¶s Construction (Example - (a|b) * a )
a: b:
a I a I I b I I I b I a I I I

b

(a | b)

I

(a|b)

*

I

(a|b) * a

I

I I

I a

I I I a

b I
24

Converting a NFA into a DFA (subset construction)
put I-closure({s0}) as an unmarked state into the set of DFA (DS) while (there is one unmarked S1 in DS) do I-closure({s0}) is the set of all states can be accessible begin from s0 by I-transition. mark S1 set of states to which there is a transition on for each input symbol a do a from a state s in S1 begin S2 I-closure(move(S1,a)) if (S2 is not in DS) then add S2 into DS as an unmarked state transfunc[S1,a] S2 end end   a state S in DS is an accepting state of DFA if a state in S is an accepting state of NFA the start state of DFA is I-closure({s0})

25

Converting a NFA into a DFA (Example)
0 I I 1 I 4 b I 5 2 a 3 I I 6 I 7 a 8

Converting a NFA into a DFA (Example ± cont.)
S0 is the start state of DFA since 0 is a member of S0={0,1,2,4,7} S1 is an accepting state of DFA since 8 is a member of S1 = {1,2,3,4,6,7,8}

a S1 a S0 b S2 b b a

27

Converting Regular Expressions Directly to DFAs
 We may convert a regular expression into a DFA (without creating a NFA first).  First we augment the given regular expression by concatenating it with a special symbol #. r (r)# augmented regular expression  Then, we create a syntax tree for this augmented regular expression.  In this syntax tree, all alphabet symbols (plus # and the empty string) in the augmented regular expression will be on the leaves, and all inner nodes will be the operators in that augmented regular expression.  Then each alphabet symbol (plus #) will be numbered (position numbers).

28

Regular Expression
(a|b) * a

DFA (cont.)

(a|b) * a # augmented regular expression

y y

Syntax tree of (a|b) * a #
# 4

* | a 1 b 2

a 3

 each symbol is numbered (positions)  each symbol is at a leave  inner nodes are operators

29

followpos
Then we define the function followpos for the positions (positions assigned to leaves). followpos(i) -- is the set of positions which can follow the position i in the strings generated by the augmented regular expression. For example, ( a | b) * a #
1 2 3 4 followpos is just defined for leaves, it is not defined for inner nodes.

firstpos, lastpos, nullable
 To evaluate followpos, we need three more functions to be defined for the nodes (not just for leaves) of the syntax tree.  firstpos(n) -- the set of the positions of the first symbols of strings generated by the sub-expression rooted by n.  lastpos(n) -- the set of the positions of the last symbols of strings generated by the sub-expression rooted by n.  nullable(n) -- true if the empty string is a member of strings generated by the sub-expression rooted by n false otherwise
31

1. If n is concatenation-node with left child c1 and right child c2, and i is a position in lastpos(c1), then all positions in firstpos(c2) are in followpos(i). 2. If n is a star-node, and i is a position in lastpos(n), then all positions in firstpos(n) are in followpos(i).  If firstpos and lastpos have been computed for each node, followpos of each position can be computed by making one depth-first traversal of the syntax tree.

 After we calculate follow positions, we are ready to create DFA for the regular expression.
34

Algorithm (RE
   

DFA)

Create the syntax tree of (r) # Calculate the functions: followpos, firstpos, lastpos, nullable Put firstpos(root) into the states of DFA as an unmarked state. while (there is an unmarked state S in the states of DFA) do ± mark S ± for each input symbol a do  let s1,...,sn are positions in S and symbols in those positions are a  S¶ followpos(s1)  ...  followpos(sn)  move(S,a) S¶  if (S¶ is not empty and not in the states of DFA) ± put S¶ into the states of DFA as an unmarked state.

 the start state of DFA is firstpos(root)  the accepting states of DFA are all states containing the position of #
35

Minimizing Number of States of a DFA
 partition the set of states into two groups:
± G1 : set of accepting states ± G2 : set of non-accepting states

 For each new group G
± partition G into subgroups such that states s1 and s2 are in the same group iff for all input symbols a, states s1 and s2 have transitions to states in the same group.

 Start state of the minimized DFA is the group containing the start state of the original DFA.  Accepting states of the minimized DFA are the groups containing the accepting states of the original DFA.

 What is the end of a token? Is there any character which marks the end of a token?
± ± ± ± ± It is normally not defined. If the number of characters in a token is fixed, in that case no problem: + But < < or <> (in Pascal) The end of an identifier : the characters cannot be in an identifier can mark the end of token. We may need a lookhead
 In Prolog: p :- X is 1. p :- X is 1.5. The dot followed by a white space character can mark the end of a number. But if that is not the case, the dot must be treated as a part of the number.

49

Some Other Issues in Lexical Analyzer (cont.)
 Skipping comments
± Normally we don¶t return a comment as a token. ± We skip a comment, and return the next token (which is not a comment) to the parser. ± So, the comments are only processed by the lexical analyzer, and the don¶t complicate the syntax of the language.

 Symbol table interface
± symbol table holds information about tokens (at least lexeme of identifiers) ± how to implement the symbol table, and what kind of operations.
 hash table ± open addressing, chaining  putting into the hash table, finding the position of a token from its lexeme.